Is it permissible to use medicine extracted from pig brain for a sick child, especially if there is no effective alternative and the doctor has confirmed that the prohibition is limited to meat only?

The physician's statement that the prohibition of pork applies only to its meat is incorrect. All parts of a pig are forbidden and impure according to the majority of scholars, even during its lifetime. All its parts are impure after its death by scholarly consensus. An exception to this is using parts of a pig for medicinal purposes if the medicine is effective and there is no pure alternative available.